[en] The concept of nonlinear normal modes (NNMs) is discussed in the present paper and its companion, Part II. Because there is virtually no application of the NNMs to large-scale engineering structures, these papers are an attempt to highlight one aspect that might drive their development in the future. Specifically, we support that numerical methods for the continuation of periodic solutions pave the way for an effective and practical computation of NNMs. In this context, we show that the NNM computation is possible with limited implementation effort. The proposed algorithm, implemented in MATLAB, relies on two main techniques, namely a shooting procedure and a method for the continuation of NNM motions. The algorithm is demonstrated using a 2DOF nonlinear system. A comparison with the results given by the AUTO software is achieved in Part II.